Abstract :
Let G be a connected reductive group acting on a finite-dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring of polynomial functions becomes a Poisson algebra. The ring of invariants is a sub-Poisson algebra. We call V multiplicity free if is Poisson commutative, i.e., if {f,g}=0 for all invariants f and g. Alternatively, G also acts on the Weyl algebra and V is multiplicity free if and only if the subalgebra of invariants is commutative. In this paper we classify all multiplicity free symplectic representations.
